Unexpected Stein fillings, rational surface singularities, and plane curve arrangements

TL;DR

This paper explores the relationship between Stein fillings and Milnor fibers of rational surface singularities, revealing new Stein fillings beyond Milnor fibers and identifying conditions for uniqueness of Stein fillings.

Contribution

It develops a symplectic approach to classify Stein fillings of links of rational surface singularities, showing existence of non-Milnor Stein fillings and conditions for their uniqueness.

Findings

Many rational singularities admit Stein fillings not diffeomorphic to Milnor fibers.

For singularities with certain resolution properties, the Stein filling is unique and given by the Milnor fiber.

The work introduces a symplectic construction using planar open books and Lefschetz fibrations.

Abstract

We compare Stein fillings and Milnor fibers for rational surface singularities with reduced fundamental cycle. Deformation theory for this class of singularities was studied by de Jong-van Straten in [dJvS98]; they associated a germ of a singular plane curve to each singularity and described Milnor fibers via deformations of this singular curve. We consider links of surface singularities, equipped with their canonical contact structures, and develop a symplectic analog of de Jong-van Straten's construction. Using planar open books and Lefschetz fibrations, we describe all Stein fillings of the links via certain arrangements of symplectic disks, related by a homotopy to the plane curve germ of the singularity. As a consequence, we show that many rational singularities in this class admit Stein fillings that are not strongly diffeomorphic to any Milnor fibers. This contrasts with previously known cases, such as simple and quotient surface singularities, where Milnor fibers are known to give rise to all Stein fillings. On the other hand, we show that if for a singularity with reduced fundamental cycle, the self-intersection of each exceptional curve is at most -5 in the minimal resolution, then the link has a unique Stein filling (given by a Milnor fiber).